Third-order Accuracy Research Articles

A Higher-Order Finite Difference Scheme for Singularly Perturbed Parabolic Problem

In this paper, we deal with a singularly perturbed parabolic convection-diffusion problem. Shishkin mesh and a hybrid third-order finite difference scheme are adopted for the spatial discretization. Uniform mesh and the backward Euler scheme are used for the temporal discretization. Furthermore, a preconditioning approach is also used to ensure uniform convergence. Numerical experiments show that the method is first-order accuracy in time and almost third-order accuracy in space.

Vertikal'nyy perenos impul'sa inertsionno-gravitatsionnymi vnutrennimi volnami na dvumernom sdvigovom techenii

Purpose. The paper is aimed at investigating the momentum vertical transfer by inertia-gravity internal waves on a two-dimensional flow with a vertical shear of velocity, and also at studying the Stokes drift of liquid particles and the mean current effect on it. Methods and Results. Free internal waves in an infinite basin of constant depth are considered in the Boussinesq approximation with the regard for the Earth rotation. Two components of the mean current velocity depend on the vertical coordinate. The equation for the vertical velocity amplitude has complex coefficients; therefore the eigenfunction and the wave frequency are complex. The corresponding boundary value problem is solved numerically by the implicit Adams scheme of the third order of accuracy. The wave frequency at a fixed wavenumber was found by the shooting method. It was determined that the frequency imaginary part was small and could be either negative or positive depending on a wave number and a mode number. Thus, both weak attenuation and weak amplification of an internal wave are possible. The vertical wave momentum fluxes are nonzero and can exceed the corresponding turbulent fluxes. The Stokes drift velocity, transverse to the wave direction, is nonzero and less than the longitudinal velocity. The vertical component of the Stokes drift velocity is also nonzero and four orders of magnitude less than the longitudinal component. The signs of the vertical component of the Stokes drift velocity for the waves with the frequencies 10 and 16 cycle/h are opposite, since the signs of their frequency imaginary parts are different; and the vertical component of the Stokes drift velocity is proportional to the wave frequency imaginary part. Conclusions. The vertical momentum wave flux of inertia-gravity internal waves differs from zero in the presence of the current whose velocity component, transverse to the wave propagation direction, depends on the vertical coordinate. The component of the Stokes drift velocity, transverse to the wave propagation direction, is nonzero and less than the longitudinal one. The vertical component of the Stokes drift velocity is also nonzero and can contribute to formation of the vertical fine structure

Vertical Transfer of Momentum by Inertia-Gravity Internal Waves on a Two-Dimensional Shear Flow

Purpose. The paper is aimed at investigating the momentum vertical transfer by inertia-gravity internal waves on a two-dimensional flow with a vertical shear of velocity, and also at studying the Stokes drift of liquid particles and the mean current effect on it. Methods and Results. Free internal waves in an infinite basin of constant depth are considered in the Boussinesq approximation with the regard for the Earth rotation. Two components of the mean current velocity depend on the vertical coordinate. The equation for the vertical velocity amplitude has complex coefficients; therefore the eigenfunction and the wave frequency are complex. The corresponding boundary value problem is solved numerically by the implicit Adams scheme of the third order of accuracy. The wave frequency at a fixed wavenumber was found by the shooting method. It was determined that the frequency imaginary part was small and could be either negative or positive depending on a wave number and a mode number. Thus, both weak attenuation and weak amplification of an internal wave are possible. The vertical wave momentum fluxes are nonzero and can exceed the corresponding turbulent fluxes. The Stokes drift velocity, transverse to the wave direction, is nonzero and less than the longitudinal velocity. The vertical component of the Stokes drift velocity is also nonzero and four orders of magnitude less than the longitudinal component. The signs of the vertical component of the Stokes drift velocity for the waves with the frequencies 10 and 16 cph are opposite, since the signs of their frequency imaginary parts are different; and the vertical component of the Stokes drift velocity is proportional to the wave frequency imaginary part. Conclusions. The vertical momentum wave flux of inertia-gravity internal waves differs from zero in the presence of the current whose velocity component, transverse to the wave propagation direction, depends on the vertical coordinate. The component of the Stokes drift velocity, transverse to the wave propagation direction, is nonzero and less than the longitudinal one. The vertical component of the Stokes drift velocity is also nonzero and can contribute to formation of the vertical fine structure.

Modeling and animation of spring dynamic absorber

The investigation of the action of the spring dynamic absorber, as well as the calculation of the physical quantities of the constructed model and animation, was carried out using non-linear methods of the Runge-Kutta type. Using only three references to the right-hand side of the differential equation, a method of the third order of accuracy is constructed, as well as two-sided approximations of the second order of accuracy. An explicit estimate of the error at each point of integration is given.

Optimal local truncation error method for solution of wave and heat equations for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes

Recently we have developed the optimal local truncation error method (OLTEM) for PDEs with constant coefficients on irregular domains and unfitted Cartesian meshes. However, many important engineering applications include domains with different material properties (e.g., different inclusions, multi-material structural components, etc.) for which this technique cannot be directly applied. In the paper OLTEM is extended to a much more general case of PDEs with discontinuous coefficients and can treat the above-mentioned applications. We show the development of OLTEM for the 1-D and 2-D scalar wave equation as well as the heat equation using compact 3-point (in the 1-D case) and 9-point (in the 2-D case) stencils that are similar to those for linear quadrilateral finite elements. Trivial unfitted Cartesian meshes are used for OLTEM with complex interfaces between different materials. The interface conditions on the interfaces where the jumps in material properties occur are added to the expression for the local truncation error and do not change the width of the stencils. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique at a given width of stencil equations. In contrast to the second order of accuracy for linear finite elements, OLTEM provides the fourth order of accuracy in the 1-D case and in the 2-D case for horizontal interfaces as well as the third order of accuracy for the general geometry of smooth interfaces. The numerical results for the domains with complex smooth interfaces show that at the same number of degrees of freedom, OLTEM is even much more accurate than quadratic finite elements and yields the results close to those for cubic finite elements with much wider stencils. The wave and heat equations are uniformly treated with OLTEM. OLTEM can be directly applied to other partial differential equations.

Compact High-Order Gas-Kinetic Scheme for Three-Dimensional Flow Simulations

A third-order compact gas-kinetic scheme is proposed for three-dimensional compressible flow computations. The new scheme is based on three key ingredients: the time-accurate gas evolution model for interface flux, the Hermite weighted essentially nonoscillatory reconstruction, and the two-stage temporal discretization. In contrast to the Riemann-solver-based methods, due to the use of time-accurate flux function the proposed scheme can achieve a third-order temporal accuracy with two stages instead of three stages in the standard Runge–Kutta method. As an extension of the existing high-order reconstructions, a Hermite weighted essentially nonoscillatory reconstruction is specifically designed for the current scheme with the implementation of the constrained least-square technique, which subsequently improves the accuracy and robustness of the compact scheme. There is no trouble cell identification in the current scheme. A third-order accuracy can be achieved even with curved boundary. Numerical examples for both smooth and discontinuous flows show the robustness and high accuracy of the compact third-order scheme, which has a comparable performance as the fifth-order noncompact gas-kinetic scheme. A large Courant–Friedrichs–Lewy number around 0.5 can be used in the computations.

An adaptive and explicit fourth order Runge–Kutta–Fehlberg method coupled with compact finite differencing for pricing American put options

We propose an adaptive and explicit Runge–Kutta–Fehlberg method coupled with a fourth-order compact scheme to solve the American put options problem. First, the free boundary problem is converted into a system of partial differential equations with a fixed domain by using logarithm transformation and taking additional derivatives. With the addition of an intermediate function with a fixed free boundary, a quadratic formula is derived to compute the velocity of the optimal exercise boundary analytically. Furthermore, we implement an extrapolation method to ensure that at least, a third-order accuracy in space is maintained at the boundary point when computing the optimal exercise boundary from its derivative. As such, it enables us to employ fourth-order spatial and temporal discretization with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary. The advantage of the Runge–Kutta–Fehlberg method is based on error control and the adjustment of the time step to maintain the error at a certain threshold. By comparing with some existing methods in the numerical experiment, it shows that the present method has a better performance in terms of computational speed and provides a more accurate solution.

Selecting the load at the intermediate time point of the ρ∞-Bathe time integration scheme

The objective of this paper is to identify the optimal load selection at the intermediate time point of the ρ∞-Bathe time integration method. We study the truncation errors of the scheme in homogeneous and forced responses for various parameters. The optimal load at the sub-step is determined by minimizing the global truncation errors of forced responses. A numerical impulse analysis shows that the optimal load at the sub-step thus established actually corresponds to numerical impulses at the three- and four-point Newton-Cotes formulas for the second- and third-order accuracy cases, respectively. We illustrate the findings of our theoretical study in example solutions of two-dimensional structural dynamics and wave propagation problems. With the optimally selected load at the sub-step, more accurate solutions can be obtained in some analyses.

An improved higher-order explicit time integration method with momentum corrector for linear and nonlinear dynamics

An improved explicit time integration method is proposed for linear and nonlinear dynamics. Its calculation procedure is obtained with cubic B-spline interpolation approximation and weighted residual method. In the formulation, a momentum corrector is used to improve actual computation accuracy, especially for some special discontinuous loads. Analytical solutions of the local truncation errors, algorithmic damping and period elongation have been deduced to obtain the influence of algorithmic parameters on these basis algorithmic properties. The proposed method possesses at least second-order accuracy and can achieve at most third-order accuracy for no physical damping case. With free algorithmic parameters, the proposed method has controllable stability and numerical dissipation. Some demonstrative numerical examples are tested to confirm high efficiency of the proposed method for a variety of dynamic problems such as, dynamic response analysis of linear systems under various representative applied loads, finite element analysis (FEA) for dynamic response of engineering structures, and nonlinear dynamic analysis for strong nonlinear system.

A study of higher-order reconstruction methods for genuinely two-dimensional Riemann solver

Balsara's genuinely multidimensional Riemann solvers, which possess a simple closed-form and high efficiency in the low-order cases, realize their multidimensional effect successfully by introducing the vertex-based framework. However, traditional higher-order (third-order and above) reconstruction methods built for the structured grids become ambiguous when applied directly into these solvers. Nowadays, there is little research on the higher-order reconstruction methods for such Riemann solvers besides Balsara's work by adopting the ADER-WENO schemes. Based on Balsara's work, we conduct research on the multidimensional higher-order reconstruction procedure in this study. This paper investigates two third-order reconstruction methods with the WENO approach for the genuinely two-dimensional HLLE Riemann solver. The critical idea of the reconstruction methods is to utilize the solution or its derivative to construct the spatial two-dimensional interpolation polynomials with the third-order accuracy. The first reconstruction method denoted by TWENO constructs the polynomials by the Taylor expansion, while the second one denoted by HWENO adopts the Hermite interpolation polynomials. By combining the polynomials with the WENO limiter, these schemes can provide required reconstructed values at the midpoints and corner-points of the cell interfaces for the framework and avoid oscillations near discontinuities. A discontinuity-detector technology is adopted to improve the computational efficiency of these reconstruction methods in simulating complex flows. Several numerical test cases are conducted and show that these third-order methods achieve their design accuracy and capture the shock waves with no overshoots or oscillations. The multidimensional nature of the solutions is kept well by the TWENO/HWENO methods, and the computational time is reduced by about 30% with the discontinuity-detector technology. Moreover, it is found that the genuinely two-dimensional Riemann solver fits the third-order reconstruction methods naturally due to its unique application of a Simpson rule.

A novel decoupling dynamic method with third-order accuracy and controllable dissipation

In this paper, an efficient explicit integration method with third-order accuracy and controllable dissipation properties is proposed, and a new accuracy criterion is presented to evaluate relative errors of integration methods. By using this method, a desirable accuracy for responses in the low-frequency domain can be attained, spurious vibrations in the high-frequency domain can be strongly suppressed. Moreover, for the proposed method, update values of the displacement and velocity do not depend on the new state. Hence, it is convenient for decoupling a multi-spring coupling dynamic system to achieve high computational efficiency. The standard formulations of the proposed method, including the second-order and third-order accuracy schemes, are derived by discussing the local truncation error, difference accuracy, and stability firstly. Subsequently, the accuracy properties of the proposed method, e.g., the period elongation and amplitude decay, are analyzed in comparison with other available state-of-the-art explicit integration methods. Additionally, the properties of the proposed method, including the accuracy, convergence, dissipation, efficiency, and nonlinearity, are evaluated by using three examples. More specifically, the accuracy and convergence are evaluated through an example with theoretical solutions; the dissipation and efficiency are investigated by analyzing a geodesic dome truss; and the effectiveness in terms of the nonlinearity is investigated via a typical nonlinear dynamic system.

A third-order subcell finite volume gas-kinetic scheme for the Euler and Navier-Stokes equations on triangular meshes

A third-order gas-kinetic scheme (GKS) based on the subcell finite volume (SCFV) method is developed for the Euler and Navier-Stokes equations on triangular meshes, in which a computational cell is subdivided into four subcells. The scheme combines the compact high-order reconstruction of the SCFV method with the high-order flux evolution of the gas-kinetic solver. Different from the original SCFV method using weighted least-square reconstruction along with a conservation correction, a constrained least-square reconstruction is adopted in SCFV-GKS so that the correction is not necessary and continuous polynomials inside each main cell can be reconstructed. For flows with large gradients, a compact hierarchical WENO limiting is adopted. In the gas-kinetic flux solver, the inviscid and viscous flux are coupled and computed simultaneously. Moreover, the spatial and temporal evolution are coupled nonlinearly which enables the developed scheme to achieve third-order accuracy in both space and time within a single stage. As a result, no quadrature point is required for the flux evaluation and the multi-stage Runge-Kutta method is unnecessary as well. A third-order k-exact finite volume GKS is also constructed with the help of k-exact reconstruction. Compared to the k-exact GKS, SCFV-GKS is compact, and with less computational cost for the reconstruction which is based on the main cell. Besides, a continuous reconstruction can be adopted in SCFV-GKS among subcells in smooth flow regions, which results in less numerical dissipation and less computational cost for flux evolution. Thus the efficiency of SCFV-GKS is much higher. Numerical tests demonstrate the strong robustness, high accuracy and efficiency of SCFV-GKS in a wide range of flow problems from nearly incompressible to supersonic flows with strong shock waves.

Third order positivity-preserving direct discontinuous Galerkin method with interface correction for chemotaxis Keller-Segel equations

In this article, we apply direct discontinuous Galerkin method with interface correction (DDGIC) to solve chemotaxis Keller-Segel equation and prove the quadratic polynomial solution satisfying positivity-preserving with third order of accuracy. We show DDGIC method can obtain optimal convergence order for the cell density variable without introducing extra auxiliary variables to approximate the gradient of the chemical concentration. This is due to the super convergent property of the DDGIC method. We prove that, with the proper choice of the numerical flux coefficients, the cell density approximation can be preserved none negative at all time. Uniform third order of accuracy is maintained with the positivity-preserving limiter applied. For the chemotaxis model with blow up or singular solutions, the DDGIC method can effectively remove negative values approximations and accurately capture the blow up time.

A fast and high-order numerical method for nonlinear fractional-order differential equations with non-singular kernel

Efficient and fast explicit methods are proposed to solve nonlinear Caputo-Fabrizio fractional differential equations, where Caputo-Fabrizio operator is a new proposed fractional derivative with a smooth kernel. The proposed methods produce the second-order for linear interpolation and the third-order accuracy for quadratic interpolation, respectively. The convergence analysis is proved by using discrete Gronwall's inequality. Furthermore, applying the recurrence relation of the memory term, it reduces CPU time executed the proposed methods. The proposed fast algorithm requires approximately O(N) arithmetic operations while O(N2) is required in case of the regular predictor-corrector schemes, where N is the total number of the time step. The following numerical examples demonstrate the accuracy of the proposed methods as well as the efficiency: nonlinear fractional differential equations, time-fraction sub-diffusion, and time-fractional advection-diffusion equation. Numerical experiments also verify the theoretical convergence rates.

Additional Consideration of the Spectral Properties of Synchrotron Light

With the help of an identified small parameter, asymptotic of the Bessel function are found here for the first time out to the third order of accuracy. On the basis of these asymptotic, spectral expressions for synchrotron radiation, suitable for physical applications, are calculated in greater detail.

The numerical solution of the problem of the contact interaction in models with gas pockets

We present the solution to the problem of detecting the gas layers in the Arctic conditions through the numerical modelling of the seismic waves spread in the heterogeneous media. The grid-characteristic method of the third order of accuracy is used in all the computations. We build the models with the different numbers of the gas layers and analyze the full wave fields in the heterogeneous media and the anomalous fields, excluding the seismic reflections from all the geological layers except for the gas layers. The results demonstrate the possibility of applying the grid-characteristic method to solving these problems. The result seismograms allow us to determine the location of the gas layers through collecting the time moments of the coming reflections from the gas layers to the seismic receivers.

The average method is much better than average

Operator splitting is a powerful method for the numerical investigation of complex time-dependent models, where the stationary (elliptic) part consists of a sum of several structurally simpler sub-operators. As an alternative to the classical splitting methods, a new splitting scheme is proposed here, the Average Method with sequential splitting. In this method, a decomposition of the original problem is sought in terms of commuting matrices. Wedemonstrate that third-order accuracy can be achieved with the Average Method. The computational performance of the method is investigated, yielding run times 1-2 orders of magnitude faster than traditional methods.

Transformation method for a nonlinear manipulator model

A continuous improvement of manipulators with a software control continues to occupy an important place in important areas of production and management. The work considers the problem of improving low-cost robotic manipulators with programmed a control. Authors achieved improved performance by using the method of polynomial transformations. The proposed a transformation method makes it possible to effectively construct a solution for a nonlinear manipulator model with a guaranteed third-order accuracy. This paper provides a detailed description of mathematical transformations to construct a solution to a nonlinear matrix equation using Lagrange equations. The experiment shows the solution to the problem of moving an object from the initial to the ultimate point of space. The results of the experiment show the change in angles of the six links of the manipulator when moving from the initial position of the grip to the final position, as well as speeds for six links of the manipulator with three sections of a movement for each link: uniformly accelerated a movement, a uniform movement with working speed and uniformly slowed down a movement to stop at the final set point. The accuracy of the solution confirmed the author’s method by the presented a method of dynamic equations. The method can solve nonlinear models.

Stabilized Integrating Factor Runge--Kutta Method and Unconditional Preservation of Maximum Bound Principle

Maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations, in the sense that the time-dependent solution of the equation with appropriate initial and boundary conditions and nonlinear operator preserves for all time a uniform pointwise bound in absolute value. It has been a challenging problem on how to design unconditionally MBP-preserving high-order accurate time-stepping schemes for these equations. In this paper, we combine the integrating factor Runge-Kutta (IFRK) method with the linear stabilization technique to develop a stabilized IFRK (sIFRK) method, and successfully derive sufficient conditions for the proposed method to preserve MBP unconditionally in the discrete setting. We then elaborate some sIFRK schemes with up to the third-order accuracy, which are proven to be unconditionally MBP-preserving by verifying these conditions. In addition, it is shown that many classic strong stability-preserving sIFRK schemes do not satisfy these conditions except the first-order one. Extensive numerical experiments are also carried out to demonstrate the performance of the proposed method.

A single-step third-order temporal discretization with Jacobian-free and Hessian-free formulations for finite difference methods

In this paper, we present an algorithmic extension of the method called the Picard Integration Formulation (PIF) that belongs to temporal updates based on the Lax-Wendroff procedure. The new extension is called the system-free (SF) approach, which furnishes ease of calculating the Jacobian and the Hessian terms necessary for third-order temporal accuracy in the original PIF method. In contrast to the analytical calculations of the Jacobian and the Hessian tensor terms in the original PIF method, our new SF approach utilizes finite difference approximations that replace the analytical calculations of the Jacobian and Hessian terms with Jacobian-free and Hessian-free approximations in a way commonly adopted in the context of iterative methods. The resulting SF approach enables our new PIF method to be a computationally efficient single-step, third-order accurate temporal scheme, whose computational performance is twice faster than the three-stage SSP-RK3 method with the same accuracy.